MAT 111 Common Final
Fall 2007
Name:
General test instructions: Show all your work on this test paper! If you solve
a problem algebraically, show all your steps. If you solve a problem by graphing on your
calculator, show a sketch of the graph, with the solution labeled. Where appropriate, round
answers to 3 decimal places.
1. Give all real and complex solutions of
x2 + 4x + 9 = 0. Simplify your answer
completely.
2. Solve
√
2x + 1 − x = −1.
3. Solve this system of equations:
4x + 3y = 6
3x − 2y = 13
4. Solve this system of equations, finding all
points of intersection.
2
x + 2y = 19
2x − y = 13
5. Solve the following inequality and express
your answer in interval notation.
x+3
≥2
2x − 5
8. Solve the following equation for x.
log7 (x) + log7 (x − 6) = 1
9. The number N of bacteria in a certain culture at time t (in hours) obeys the law of
uninhibited growth N (t) = 400e0.025t .
(a) What is the population after 5 hours?
6. Write the following as a single logarithm.
4 log(x) +
1
log(x − 2) − 2 log(x + 1)
3
(b) When will the number of bacteria double?
10. A 20 lb bag of Economy brand cement contains 25% cement and 75% sand. How
much pure cement must be added to produce a cement mix that is 60% cement?
7. Solve the following equation for x.
e(5x−1) + 3 = 4
11. The function f (x) is graphed below.
12. Let f (x) =
2x+7
.
9+4x
Find f −1 (x).
Graph the following transformations
of f (x):
(a) f (x − 1) + 3
13. (a) Graph f (x) = x3 −3x2 +2x+1. Clearly
label all intercepts.
6
5
-
−5
5
−5
(b) −2f (x)
(b) f has a relative minimum of
at x =
.
(c) f has a relative maximum of
at x =
.
(d) Name all intervals on which f is increasing.
14. The revenue in dollars from selling x
units of a certain product is given by
R(x) = 25x − .05x2 .
17. Find all the zeros and their multiplicities for the polynomial function
f (x) = −2x5 + 12x4 − 18x3 .
(a) What quantity x maximizes revenue?
(b) What is the maximum revenue?
15. Divide these complex numbers, simplify,
and write the answer in standard form.
18. A polynomial with degree 3 has zeros at
4,1,-2, and its leading coefficient is negative. Sketch the graph of this polynomial.
4 + 2i
3−i
6
5
-
−5
5
−5
19. Solve |4x − 7| − 5 > −3.
16. Find an equation of the line passing
through the point (1, 3) that is perpendicular to the line y = 2x + 3.
20. Analyze the graph of y =
2x−1
.
x+5
21. Let


−5 ≤ x < −2
3
g(x) = −3x + 1 −2 ≤ x ≤ 0

−x2
x>0
(a) Find the domain.
(a) Evaluate g(−2).
(b) Find any x-intercepts.
(b) Evaluate g(−4) − 6g(3).
(c) Find any y-intercepts.
(c) Determine the domain of g(x).
(d) Find any vertical asymptotes.
(d) Graph g(x).
(e) Find any horizontal asymptotes.
6
5
(f) Sketch a graph of the function.
-
6
−5
5
-
−5
5
−5
5
−5
(e) Find the range of g(x).
22. Let f (x) = x21−2 and g(x) =
(a) Find (f ◦ g)(x).
√
x − 3.
24. Solve 5 · 3x = 50. Provide both an exact
answer and a decimal approximation.
(b) Find the domain of (f ◦ g)(x).
√
25. Solve x + 5 x − 14 = 0.
23. Susan is planning to retire in 20 years. She
calculates that she will need $100,000 in
addition to the money she will get from her
retirement plans. How much should she invest today at 10% compounded weekly to
accomplish her objective?
26. The United States gross national product for the
years 1998-2002, in trillions of dollars, is given in
the table below, with year 1 corresponding to 1998.
Year
1
2
3
4
Amount in Account 8.78 9.30 9.85 10.10
(a) Use your graphing calculator to fit a linear
model to the data. Write the equation and
correlation coefficient here.
(b) Use your graphing calculator to fit an exponential model to the data. Write the equation
and correlation coefficient here.
(c) Which equation is a better model of the data?
Explain.
5
10.44